The 
th
complex Fourier coefficient of a real- or complex-valued periodic
function 
with period 
, Lebesgue integrable
over one period, is
 |
where 
is an integer, 
is a real number, and 
is any real
number. The integrand is 
-periodic, so the value is independent of 
. The Fourier coefficients are expansion
coefficients in the corresponding Fourier series
of 
. Their existence does not imply that the series converges pointwise to 
(Axler 2026, p. 342).
In the Wolfram Language, FourierCoefficient[expr, t, n] implements complex Fourier coefficients with default period 
. The separate half-range Fourier
cosine series and Fourier sine series on
![[0,pi]](/images/equations/FourierCoefficient/Inline12.svg)
have their coefficients implemented as FourierCosCoefficient[expr,
t, n] and FourierSinCoefficient[expr,
t, n], respectively.
